Pi and irrational numbers

[MPD-MathDummy-Guy]

I don’t know where to post my question and am agog that it has not been previously asked. I’m trying to find the formula used to calculate pi to one million non-repeating places places. The “pi formula” (circumference/diameter) always requires I use two finite numbers which always result in a finite or repeating answer. There is no way to measure circumference and diameter sufficiently accurately to produce one million non-repeating decimal places or more. What formula is used that mathematics jumps to the conclusion pi is an irrational number? Everything I try says it’s completely rational.

 

[lev888]

I don’t know where to post my question and am agog that it has not been previously asked. I’m trying to find the formula used to calculate pi to one million non-repeating places places. The “pi formula” (circumference/diameter) always requires I use two finite numbers which always result in a finite or repeating answer. There is no way to measure circumference and diameter sufficiently accurately to produce one million non-repeating decimal places or more. What formula is used that mathematics jumps to the conclusion pi is an irrational number? Everything I try says it’s completely rational.

Proof that π is irrational - Wikipedia

en.m.wikipedia.org

 

[Otis]

non-repeating places places

That's funny.

By the way, you can download 50 billion decimal places of pi from the US government (if you have a monster-sized hard drive). For a million digits, see this thread.

 

[JeffM]

I don’t know where to post my question and am agog that it has not been previously asked. I’m trying to find the formula used to calculate pi to one million non-repeating places places. The “pi formula” (circumference/diameter) always requires I use two finite numbers which always result in a finite or repeating answer. There is no way to measure circumference and diameter sufficiently accurately to produce one million non-repeating decimal places or more. What formula is used that mathematics jumps to the conclusion pi is an irrational number? Everything I try says it’s completely rational.

First, the ontological nature of irrational numbers has been discussed on this site. For a recent example, see

Pi

pi can be clearly "expressed" by "the ratio of the circumference of any circle divided by the radius of that circle." Yes, but those circles must be perfect, in order to express pi (not approximately, but as pi). I read Jeff's comments to mean that we cannot generate pi by anything we could...

www.freemathhelp.com

That is, one way to think about the ontology of irrational numbers is that they exist as mental constructs but do not represent anything in the physical world. (This may be a minority view among mathematicians.)

Second, mathematicians define [imath]\pi[/imath] in multiple ways. See

Pi - Wikipedia

en.wikipedia.org

A way that seems appropriate to your question is

[math]\pi = \sum_{j=0}^{\infty} (-1)^j * \dfrac{4}{2j+1}.[/math]
A computer can spit out approximations of increasing accuracy until the cows come home. Do 5000 of them, and you get

[math]3.1412 < \pi < 3.1418.[/math]

 

[pka]

I don’t know where to post my question and am agog that it has not been previously asked. I’m trying to find the formula used to calculate pi to one million non-repeating places places. The “pi formula” (circumference/diameter) always requires I use two finite numbers which always result in a finite or repeating answer. There is no way to measure circumference and diameter sufficiently accurately to produce one million non-repeating decimal places or more. What formula is used that mathematics jumps to the conclusion pi is an irrational number? Everything I try says it’s completely rational.

I have a book for you to read: THE NUMBER SENSE sub titled How the Mind Creates Mathematics by Stanislas Debaene.
Stan is a mathematician who has turned himself into a brain scientist. He is a research affiliate at the University of Paris.

 

[MPD-MathDummy-Guy]

I have a book for you to read: THE NUMBER SENSE sub titled How the Mind Creates Mathematics by Stanislas Debaene.
Stan is a mathematician who has turned himself into a brain scientist. He is a research affiliate at the University of Paris.

Thank you, I have ordered a copy from my local library.

 

[MPD-MathDummy-Guy]

First, the ontological nature of irrational numbers has been discussed on this site. For a recent example, see

Pi

pi can be clearly "expressed" by "the ratio of the circumference of any circle divided by the radius of that circle." Yes, but those circles must be perfect, in order to express pi (not approximately, but as pi). I read Jeff's comments to mean that we cannot generate pi by anything we could...

www.freemathhelp.com

That is, one way to think about the ontology of irrational numbers is that they exist as mental constructs but do not represent anything in the physical world. (This may be a minority view among mathematicians.)

Second, mathematicians define [imath]\pi[/imath] in multiple ways. See

Pi - Wikipedia

en.wikipedia.org

A way that seems appropriate to your question is

[math]\pi = \sum_{j=0}^{\infty} (-1)^j * \dfrac{4}{2j+1}.[/math]
A computer can spit out approximations of increasing accuracy until the cows come home. Do 5000 of them, and you get

[math]3.1412 < \pi < 3.1418.[/math]

Perhaps if I ask my question more directly I could get a more reasonable answer. PI has been widely pondered for centuries, if not millennia. It seems mathematicians routinely accept it is an irrational number without end. However, some formula is used by mathematicians who declare they have calculated PI to x number of places. I am trying to prove PI is not irrational, nor is any other number irrational if it is real. I believe mathematics has created an insoluble problem due to logical fallacies created by the requesters, e.g., “Can God make a mountain He can’t move”. A paradox that can’t be resolved because language rules prohibit an answer expressible in any language. Is 1/3 repeating forever without end? Of course not. Everyone knows we can divide anything into three perfectly equal pieces, however, we have no way to express it because mathematics is expressed in base 10 and decimally there is no way to express it. I am positing PI suffers from exactly the same issue, thus I want to try and solve PI to its finite and exact value using an exact and agreed to formula/methodology, but in hexadecimal

does anyone know what formula I would use?

 

[JeffM]

Perhaps if I ask my question more directly I could get a more reasonable answer. PI has been widely pondered for centuries, if not millennia. It seems mathematicians routinely accept it is an irrational number without end. However, some formula is used by mathematicians who declare they have calculated PI to x number of places. I am trying to prove PI is not irrational, nor is any other number irrational if it is real. I believe mathematics has created an insoluble problem due to logical fallacies created by the requesters, e.g., “Can God make a mountain He can’t move”. A paradox that can’t be resolved because language rules prohibit an answer expressible in any language. Is 1/3 repeating forever without end? Of course not. Everyone knows we can divide anything into three perfectly equal pieces, however, we have no way to express it because mathematics is expressed in base 10 and decimally there is no way to express it. I am positing PI suffers from exactly the same issue, thus I want to try and solve PI to its finite and exact value using an exact and agreed to formula/methodology, but in hexadecimal

does anyone know what formula I would use?

I am sorry you have found my answer unreasonable. Of course, the topic is irrational numbers so maybe an unreasonable answer is to be expected.

Mathematics is not about what you personally posit or what you personally believe; it is about what can be proved. It has been proved that [imath]\pi[/imath] is not a rational number. So your goal is impossible to achieve.

An irrational number is any finite quantity that cannot be expressed as a fraction of two integers. But every number that can be expressed as either a terminating or repeating DECIMAL is a fraction of two integers. Therefore, an irrational number cannot be expressed as a terminating or repeating decimal. That is basic logic.

You might wonder whether (if we chose a base other than ten) a different base would solve the problem. The answer is NO. It does however answer your question about one third. In base 12, 1/3 would equal 0.4. In other words, some fractions are repeating in one base but not in others. 1/3 is a rational number so there are many conceivable systems of numeration where 1/3 is expressed in a terminating form. The three systems of numeration most used in the modern world, however, do not express 1/3 in a repeating form.

The proof that [imath]\pi[/imath] is not a fraction formed from two integers took millennia to establish. It is hard. The proof that the square root of 2 is irrational is elegant and is about 2500 years old. It does not take much work to understand it. The first good system for finding an approximation for [imath]\pi[/imath] is also about 2500 years old. That method likewise is not hard to understand. So your original comment that you were agog that no one has considered these questions is simply uninformed.

 

[lev888]

Perhaps if I ask my question more directly I could get a more reasonable answer. PI has been widely pondered for centuries, if not millennia. It seems mathematicians routinely accept it is an irrational number without end. However, some formula is used by mathematicians who declare they have calculated PI to x number of places. I am trying to prove PI is not irrational, nor is any other number irrational if it is real. I believe mathematics has created an insoluble problem due to logical fallacies created by the requesters, e.g., “Can God make a mountain He can’t move”. A paradox that can’t be resolved because language rules prohibit an answer expressible in any language. Is 1/3 repeating forever without end? Of course not. Everyone knows we can divide anything into three perfectly equal pieces, however, we have no way to express it because mathematics is expressed in base 10 and decimally there is no way to express it. I am positing PI suffers from exactly the same issue, thus I want to try and solve PI to its finite and exact value using an exact and agreed to formula/methodology, but in hexadecimal

does anyone know what formula I would use?

If you run into a stone wall using a more direct route you will not get a more reasonable result. The proposition that PI is irrational has been proven, it's not just being "accepted" on faith.

 

[lookagain]

Everything I try says it’s completely rational.

We don't have samples/examples here typed from of "everything [you have tried]," so we cannot see your thought pattern to know if
you are thinking rationally about pi.

"There is no way that ..." <---- If you're not knowledgeable enough about the subject, I don't see how you can make this absolute statement.

 

[Otis]

Everyone knows we can divide anything into three perfectly equal pieces

Hi MPD-MD-G. I'm curious. How would you determine after such divisions that your three pieces are perfectly equal? For example, if you were to desire exactly one third of a dollar in cash, then you would need to cut a penny into three perfectly-equal pieces. How would you obtain/confirm exactly one third of a penny, interpreting from a mathematical point-of-view?

 

[Deleted member 4993]

Perhaps if I ask my question more directly I could get a more reasonable answer. PI has been widely pondered for centuries, if not millennia. It seems mathematicians routinely accept it is an irrational number without end. However, some formula is used by mathematicians who declare they have calculated PI to x number of places. I am trying to prove PI is not irrational, nor is any other number irrational if it is real. I believe mathematics has created an insoluble problem due to logical fallacies created by the requesters, e.g., “Can God make a mountain He can’t move”. A paradox that can’t be resolved because language rules prohibit an answer expressible in any language. Is 1/3 repeating forever without end? Of course not. Everyone knows we can divide anything into three perfectly equal pieces, however, we have no way to express it because mathematics is expressed in base 10 and decimally there is no way to express it. I am positing PI suffers from exactly the same issue, thus I want to try and solve PI to its finite and exact value using an exact and agreed to formula/methodology, but in hexadecimal

does anyone know what formula I would use?

Do you know how to prove \(\displaystyle \sqrt{2} \) is irrational?

Some Greek guy proved that ~3000 years ago (response #8)

No body yet , has found a flaw in that " proof" - thus irrational numbers "really" do exist!!!